The problems are in their interpretation and in the interpretation of much of the formalism of quantum mechanics and quantum field theory.
There are far more possible interpretations that match the known experimental facts than it would appear from university textbooks.
The problem is that for now it is not possible to decide which of them is right, because nobody has been able to work out their complete consequences by creating a complete coherent quantum theory.
There are a lot of alternative interpretations that sound more plausible than the mainstream interpretation, but none of them has been developed enough to be able to prove that they could explain everything.
This is a subject that requires a book-length treatment. There are actually some such books, but they only show the very serious flaws of the mainstream interpretation of quantum theory, without being able to propose an adequate complete replacement. Criticism is always easier than discovering a better solution.
The only certitude is that the current quantum mechanics/quantum field theory are not the final theories of physics and whoever believes that physics is solved is as naive as those who believed the same thing at the end of the 19th century.